Introduction to Mathematical Physics/Statistical physics/Canonical distribution in classical mechanics

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Consider a system for which only the energy is fixed. Probability for this system to be in a quantum state of energy is given (see previous section) by:

Consider a classical description of this same system. For instance, consider a system constituted by particles whose position and momentum are noted and , described by the classical hamiltonian . A classical probability density is defined by:

eqdensiprobaclas

Quantity represents the probability for the system to be in the phase space volume between hyperplanes and . Normalization coefficients and are proportional.

One can show [ph:physt:Diu89] that

being a sort of quantum state volume.

Remark:

This quantum state volume corresponds to the minimal precision allowed in the phase space from the Heisenberg uncertainty principle:

\index{Heisenberg uncertainty principle}

Partition function provided by a classical approach becomes thus:

But this passage technique from quantum description to classical description creates some compatibility problems. For instance, in quantum mechanics, there exist a postulate allowing to treat the case of a set of identical particles. Direct application of formula of equation eqdensiprobaclas leads to wrong results (Gibbs paradox). In a classical treatment of set of identical particles, a postulate has to be artificially added to the other statistical mechanics postulates:

Postulate:

Two states that does not differ by permutations are not considered as different.

This leads to the classical partition function for a system of identical particles: